A141095 -id:A141095 - cp's OEIS frontend

A141100 Number of unordered pairs of odd composite numbers that sum to 2n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 3, 1, 1, 3, 2, 1, 4, 1, 2, 5, 1, 3, 5, 1, 4, 5, 3, 3, 6, 3, 3, 7, 3, 3, 9, 3, 4, 7, 4, 6, 9, 5, 5, 8, 6, 6, 10, 5, 5, 12, 4, 6, 12, 5, 9, 11, 7, 7, 11, 9, 9, 13, 8, 8, 16, 7, 11, 14, 8, 11, 14, 9, 9, 17, 13, 10, 16, 11, 11, 19, 11, 12, 18, 10

Offset: 1

T. D. Noe, Jun 02 2008, Jun 05 2008

nonn

See A141099 for pairs of odd nonprime numbers. We have a(n) > 0 except for the 14 values of 2n given in A118081.

			a(18)=2 because 36 = 9+27 = 15+21.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A005171, A118081, A141095, A141097, A141099.

Table[cnt=0; Do[If[ !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,3,n,2}]; cnt, {n,100}]

a(n) = 1 - floor(n/2) + Sum_{i=3..n} c(i) * c(2n-i), n>1, where c = A005171. - Wesley Ivan Hurt, Dec 27 2013

A141097 Number of unordered pairs of coprime composite numbers that sum to 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 1, 0, 0, 3, 3, 0, 0, 3, 0, 3, 2, 0, 3, 4, 0, 1, 2, 1, 4, 5, 0, 0, 3, 1, 5, 5, 0, 4, 6, 0, 5, 2, 0, 7, 6, 0, 0, 9, 2, 8, 8, 0, 6, 4, 1, 8, 4, 1, 9, 9, 1, 4, 10, 2, 8, 11, 0, 3, 11, 3, 10, 4, 2, 12, 8, 1, 4, 13, 2

Offset: 1

T. D. Noe, Jun 02 2008

nonn

See A141095 for pairs of coprime nonprime numbers. It appears that a(n) > 0 except for the 43 values of 2n given in A141098. Roberts says that A. M. Vaidya proved that a(n) > 0 for all sufficiently large n.

			a(17)=1 because 34 = 9+25.

Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 190.

T. D. Noe, Table of n, a(n) for n=1..10000
Carlos Rivera, Conjecture 33: The Goldbach Temptation

Cf. A141099, A141100.

Table[cnt=0; Do[If[GCD[2n-i,i]==1 && !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,3,n,2}]; cnt, {n,100}]

A141099 Number of unordered pairs of odd nonprime numbers that sum to 2n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 3, 0, 2, 3, 1, 2, 3, 3, 2, 4, 2, 3, 5, 1, 4, 6, 1, 5, 5, 3, 4, 7, 3, 4, 7, 4, 4, 9, 4, 5, 8, 4, 7, 9, 5, 6, 8, 6, 7, 10, 6, 6, 13, 5, 7, 13, 5, 10, 11, 8, 8, 11, 9, 10, 14, 9, 9, 16, 7, 12, 15, 8, 12, 15, 9, 10, 17, 14, 11, 16, 12, 12, 19, 11, 13, 19

Offset: 1

T. D. Noe, Jun 02 2008, Jun 05 2008

nonn

See A141100 for pairs of odd composite numbers. We have a(n) > 0 except for the 8 values of 2n given in A046458.

			a(18)=3 because 36 = 1+35 = 9+27 = 15+21.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A141095, A141097.

Table[cnt=0; Do[If[ !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,1,n,2}]; cnt, {n,100}]

A141096 Even numbers not representable as the sum of two coprime nonprime numbers.

Original entry on oeis.org

4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 48, 54, 60, 72, 80, 84, 90, 108, 110, 132, 138, 140, 150, 180

Offset: 1

T. D. Noe, Jun 02 2008

Numbers k such that A141095(k/2) = 0.
180 is the last term.
This sequence is a subsequence of A141098.

Cf. A141095, A141098.

t = Table[Length[Select[Range[n/2], ! PrimeQ[#] && ! PrimeQ[n - #] && GCD[#, n - #] == 1 &]], {n, 2, 2000, 2}]; Flatten[2*Position[t, 0]] (* T. D. Noe, Dec 05 2013 *)

A185279 a(n) = number of ways that one can write n as the sum of two positive integers such that i) the integers are relatively prime to n but ii) the integers are not themselves prime.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 3, 0, 1, 1, 3, 0, 3, 1, 1, 1, 5, 0, 6, 0, 2, 2, 3, 1, 7, 0, 3, 1, 8, 0, 9, 1, 1, 2, 9, 0, 8, 1, 3, 2, 11, 0, 7, 1, 4, 3, 13, 0, 14, 1, 3, 4, 8, 1, 15, 1, 6, 1, 16, 0, 17, 3, 2, 4, 11, 1, 18, 0, 7, 4, 19, 0

Offset: 1

Jason Holland, Feb 19 2011

nonn

These might be called "relative Goldbach partitions."
This sequence was first discovered by my student Houston Hutchinson.
We became interested in this sequence when looking at Goldbach Partitions thus at first we only considered the even numbered terms.  The graph of the even values of a(n) looks like Goldbach's comet except with an exponential appearance rather than a logarithmic appearance.  We give a formula for the even values in the formula section.
Sequence A141095 has the terms for even n.

			a(34) is the first even term with value greater than 1.  The number 34 = 33 + 1 and 25 + 9.  The latter sums meet the requirements listed in the definition. For odd n greater than 3, a(n) will always be at least 1 since 1 + (n - 1) is a sum that satisfies the definition.  For example a(5) = 1 since 5 = 1 + 4.

T. D. Noe, Table of n, a(n) for n = 1..5000

Cf. A005171, A141095.

Table[Length[Select[Range[n/2], ! PrimeQ[#] && ! PrimeQ[n - #] && GCD[#, n - #] == 1 &]], {n, 100}] (* T. D. Noe, Dec 05 2013 *)

def A185279(n):
    return sum(1 for i in (1..n//2) if all(gcd(j,n) == 1 and not is_prime(j) for j in (i, n-i))) # D. S. McNeil, Mar 05 2011

For even n >= 4, denote the number of Goldbach partitions that have distinct primes by g(n), denote the totient of n by t(n), and denote the primes less than n that are NOT factors of n by p(n). Then a(n) = g(n)- p(n) + t(n)/2.

a(n) = Sum_{i=1..floor(n/2)} [GCD(i, n-i) = 1] * c(i) * c(n-i), where c is the characteristic function of nonprimes (A005171) and [ ] is the Iverson bracket. - Wesley Ivan Hurt, Dec 08 2020

A232721 Numbers not representable as the sum of two coprime nonprime numbers.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 48, 54, 60, 72, 80, 84, 90, 108, 110, 132, 138, 140, 150, 180

Offset: 1

Irina Gerasimova, Nov 28 2013

Numbers n such that A185279(n) = 0. 1 and 3 together with A141096.

Cf. A062303, A018252, A141095, A141098.

t = Table[Length[Select[Range[n/2], ! PrimeQ[#] && ! PrimeQ[n - #] && GCD[#, n - #] == 1 &]], {n, 2000}]; Flatten[Position[t, 0]] (* T. D. Noe, Dec 05 2013 *)

A363992 The number of ways 2n can be expressed as the sum of an odd prime number and an odd nonprime, both of which are relatively prime to n.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 3, 3, 1, 6, 3, 1, 8, 4, 2, 6, 6, 3, 5, 7, 4, 8, 8, 2, 12, 7, 3, 13, 6, 6, 11, 9, 4, 12, 12, 4, 13, 13, 3, 14, 14, 8, 17, 11, 7, 15, 15, 10, 14, 13, 7, 16, 18, 3, 22, 18, 7, 24, 14, 11, 20, 20, 14, 17, 18, 10, 22, 22, 8

Offset: 0

Brian Darrow, Jr., Jun 30 2023

nonn

			For n=24 (2n=48), we have a(24)=3 since 48=1+47, 48=13+35, and 48=23+25. These are the only sums containing one prime and one nonprime, both of which are relatively prime to n.

Cf. A002375, A141095.

f:= proc(n) local k;
   nops(select(k -> igcd(n,k) = 1 and igcd(n,2*n-k) = 1 and isprime(k) and not isprime(2*n-k), [seq(k,k=1..2*n-1,2)]))
end proc:
map(f, [$0..100]); # Robert Israel, Jul 03 2023

def d(a):
    """
    This function returns the number of ways n=2a can be expressed as the sum of one prime number and an odd composite that are relatively prime to n
    """
    d=0
    for i in range(1,a+1):
        if ((is_prime(i) and not is_prime(2*a-i) and gcd(i,2*a-i) == 1)) or ((not is_prime(i) and is_prime(2*a-i) and gcd(i,2*a-i) == 1)):
            d=d+1
    return d

cp's OEIS Frontend

A141100 Number of unordered pairs of odd composite numbers that sum to 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A141097 Number of unordered pairs of coprime composite numbers that sum to 2n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A141099 Number of unordered pairs of odd nonprime numbers that sum to 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A141096 Even numbers not representable as the sum of two coprime nonprime numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A185279 a(n) = number of ways that one can write n as the sum of two positive integers such that i) the integers are relatively prime to n but ii) the integers are not themselves prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A232721 Numbers not representable as the sum of two coprime nonprime numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A363992 The number of ways 2n can be expressed as the sum of an odd prime number and an odd nonprime, both of which are relatively prime to n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Sage